# How do you find the integral of #(e^x)(cosx) dx#?

I found:

I tried Integration by Parts (twice) and a little trick...!

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Alternatively, we can use a nice little technique called complexifying the integral.

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To find the integral of ( e^x \cdot \cos(x) , dx ), you can use integration by parts. Let ( u = e^x ) and ( dv = \cos(x) , dx ). Then, ( du = e^x , dx ) and ( v = \sin(x) ). Using the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substituting the values:

[ \int e^x \cdot \cos(x) , dx = e^x \sin(x) - \int \sin(x) \cdot e^x , dx ]

This results in another integral similar to the original one. You can solve it by using integration by parts again, or you can recognize that the integral of ( e^x \sin(x) ) can be found by using integration by parts in a similar manner as before. After solving, you'll find the antiderivative of ( e^x \cos(x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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